A Novel Approach for Improving Reverse Osmosis Model Accuracy: Numerical Optimization for Water Purification Systems

The primary objective of this study is to present a new technique and library designed to validate the outcomes of numerical methods used for addressing various issues. This paper specifically examines the reverse osmosis (RO) model, a well-known water purification system. A crucial aspect of this problem involves solving an integral that is part of the overall solution. This integral is handled using one of the quadrature integration methods, with a focus on Romberg integration in this study. To manage the number of iterations, as well as to ensure accuracy and minimize errors, we employ the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) alongside the CADNA (Control of Accuracy and Debugging for Numerical Applications) library. By implementing this approach, we aim to achieve not only optimal results, but also the best method step and minimal error, and we aim to address numerical instabilities. The results show that only 16 iterations of the Romberg integration rule will be enough to find the approximate solutions.To demonstrate the efficacy and precision of our proposed method, we conducted two comprehensive comparative studies with the Sinc integration. The first study compares the optimal iteration count, optimal approximation, and optimal error between the single and double exponential decay methods and the Romberg integration technique. The second study evaluates the number of iterations required for convergence within various predefined tolerance values. The findings from both studies consistently indicate that our method outperforms the Sinc integration in terms of computational efficiency. Additionally, these comparative analyses highlight the potential of our approach as a reliable and effective tool for numerical integration.